Quantitative Aptitude Notes Part 1(SIMPLIFICATION)

chapter-1

SIMPLIFICATION

 

BASIC FORMULAE

  1. (a+b)2=a2+b2+2ab
  2. (a−b)2=a2+b2−2ab
  3. (a+b)2−(a−b)2=4ab
  4. (a+b)2+(a−b)2=2(a2+b2)
  5. (a2–b2)=(a+b)(a−b)
  6. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
  7. (a3+b3)=(a+b)(a2−ab+b2)
  8. (a3–b3)=(a−b)(a2+ab+b2)
  9. (a3+b3+c3−3abc)=(a+b+c)(a2+b2+c2−ab−bc−ca)
  10. If a+b+c=0, then a3+b3+c3=3abc.

 

 

TYPES OF NUMBERS

 

 

  1. Natural Numbers:

Counting numbers 1,2,3,4, 5 … are called natural numbers

 

  1. Whole Numbers:

All counting numbers together with zero form the set of whole numbers.
Thus,
(I) 0 is the only whole number which is not a natural number.
(II) Every natural number is a whole number.

 

 

 

 

  1. Integers:

All  natural  numbers,  0  and  negatives  of  counting  numbers i.e.,…,−3,−2,−1,0,1,2,3,….. together form the set of integers.
(i) Positive Integers: 1,2,3, 4….. is the set of all positive integers.
(ii) Negative Integers: −1,−2,−3… is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
So,  0,1,2,3,….  represents  the  set  of  non-negative  integers,
while 0,−1,−2,−3,….. represents the set of non-positive integers.

 

  1. Even Numbers:

A number divisible by 2 is called an even number, ex. 2, 4, 6, 8, etc.

 

  1. Odd Numbers:

A number not divisible by 2 is called an odd number. e.g. 1, 3,5,7,9, 11 etc.

 

  1. Prime Numbers:

A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

 

  1. Composite Numbers:

Numbers  greater  than  1  which  are  not  prime,  are  known  as composite numbers, e.g., 4,6,8,9,10,12.
Note:
(i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.

REMAINDER AND QUOTIENT:
“The remainder is r when p is divided by k” means p=kq+r the integer q is called the quotient.
EVEN ,ODD NUMBERS
A number n is even if the remainder is zero when n is divided by 2: n=2z+ 0 or n=2z.
A number n is odd if the remainder is one when n is divided by 2: n=2z+1.
even X even = even
odd X odd = odd
even X odd = even
even + even = even
odd + odd = even
even + odd = odd

 

 

 

Some important tricks

  1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
  2. (12+ 22 + 32 + ….. + n2) = n ( n + 1 ) (2n + 1) / 6
  3. (13+ 23 + 33 + ….. + n3) = (n(n + 1)/ 2)2
  4. Sum of first n odd numbers = n2
  5. Sum of first n even numbers = n (n + 1)

For square and square root or cube and cube root

You must learn square and cube at least from number 1 to 50

Short tricks of multiplication

 

Multiplication of Two digit numbers:
(1)13 x 13 = ?
The result of multiplication of two digit number is 13×13 = 169.
Step 1: Multiply 3×3 = 9 then,
Step 2: Do Cross-multiplication (1×3) = 3 and (1×3) = 3.
Step 3: Add both the result (1×3 + 1×3) = 6 and write down to the left of 9 (result of step 1).
Step 4: Multiply left hand side numbers (1×1) = 1 and write down to the left of 6 (result of step 3).
Finally the result we get 169. (Try to calculate all four steps in mind.)

 

 

(2) 87 x 33 =?
The result of multiplication of two digit number is 88 x 33 = 2871
Step 1: Multiply (3×7) = 21 note down 1 and carry 2 then,
Step 2: Do Cross-multiplication (3×8) = 24 and (3×7) = 21.
Step 3: Add both the result with carry (24 + 21 + 2) = 47 and write down 7 carry 4.
Step 4: Multiply left hand side numbers (3×8) = 24 and add carry (24 + 4) write down to the left of 7
finally the result we get 2871. (Try to calculate all four steps in mind.)

Multiplication of a Three digit numbers
(1) 175 x 157 =?
The result of multiplication of three digit number is 175×157 = 27475.
Step 1: Multiply (5×7) = 35 (note down 5 carry 3).
Step 2: Then do cross multiplication (7×7 + 5×5 + 3 (add carry)) = 77 (note down 7 carry 7).
Step 3: Again (1×7 + 1×5 + 7×5 + 7 (add carry)) = 54 (note down 4 carry 5).
Step 4: do cross multiplication and add carry (1×5 + 1×7 + 5 (add carry)) = 17 (note down 7 carry 1).
Step 5: Again (1×1 + 1) = 2, note it down.
And finally the result we get 27475.
(2) 275×354 =?
The result of multiplication of three digit number is 275×354 = 97350.
Step 1: Multiply (4×5) = 20 (note down 0 carry 2).
Step 2: Then do cross multiplication (5×5 + 4×7 + 2 (add carry)) = 55 (note down 5 carry 5).
Step 3: Again (4×2 + 3×5 + 5×7 + 5 (add carry)) = 63 (note down 3 carry 6).
Step 4: Again do cross multiplication and add carry (5×2 + 3×7 + 6) = 37 (note down 7 carry 3).
Step 5: do multiplication of left numbers and add carry  (3×2 + 3) = 9, note it down.
And finally the result we get 97350.

 

 

Multiplication of Three and Two digit numbers
(1) 295 x 19 =?
The result of multiplication of three and two digit number is 295×19 = 5605.
Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).
Step 2: Then do cross multiplication and add carry (9×9 + 5×1 + 4) = 90 (note down 0 and carry 9).
Step 3: Again do cross multiplication and add carry (2×9 + 1×9 +9) = 36 (note down 6 and carry 3).
Step 4: Now multiply of left numbers and add carry (2×1 + 3) = 5, note it down.
And finally the result we get 5605.

 

(2) 195 x 19 =?
The result of multiplication of three and two digit number is 195 x 19 = 3705.
Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).
Step 2: Then do cross multiplication and add carry (9×9 + 5×1 + 4) = 90 (note down 0 and carry 9).
Step 3: Again do cross multiplication and add carry (1×9 + 1×9 +9) = 27 (note down 7 and carry 2).
Step 4: Now multiply of left numbers and add carry (1×1 + 2) = 3, note it down.
And finally the result we get 3705.

Multiplication of Four and Two digit numbers

(1)4295 x 19 =?
The result of multiplication of three and two digit number is 4295×19 = 81605.
Step 1: Multiply 5×9 = 45 (note down 5 and carry 4).
Step 2: Then do cross multiplication and add carry (1×9 + 9×9 + 4) = 90 (note down and carry 9).
 

Step 3: Again do cross multiplication and add carry (1×9 + 9×2 +9) = 36 (note down 6 and carry 3).
Step 4: Again do cross multiplication and add carry (9×4 + 1×2 +3) =41 (note down 1 and carry 4).
Step 5: Now multiply of left numbers and add carry (1×4 + 4) = 8, note it down.
And finally the result we get 81605.
(2) 3457 x 23 =?
The result of multiplication of three and two digit number is 4295×19 = 79511.
Step 1: Multiply 3×7 = 21 (note down 1 and carry 2).
Step 2: Then do cross multiplication and add carry (3×5 + 2×7 + 2) = 31 (note down and carry 3).
Step 3: Again do cross multiplication and add carry (2×5 + 3×4 +3) = 25 (note down and carry 2).
Step 4: Again do cross multiplication and add carry (2×4 + 3×3 +3) = 19 (note down 9 and carry 1).
Step 5: Now multiply of left numbers and add carry (2×3 + 1) = 7, note it down.
And finally the result we get 79511.